Answer
verify the identity
Work Step by Step
$sin^{2}θ$ + $cos^{2}θ$=1
(a) $tan θ sin θ + cos θ = sec θ$
left:
$tan θ sin θ + cos θ$
=$\frac{sin θ}{cos θ} sin θ +cos θ$
=$\frac{sin^{2} θ}{cos θ} +cos θ$
=$\frac{sin^{2} θ + cos^{2}θ}{cos θ}$
=$\frac{1}{cosθ}$
=$sec θ$
left=right
so, proved
$tan x =\frac{sin x}{cos x}$
(b) $\frac{2tan x}{1+tan^{2}x}=sin 2x$
left:
$\frac{2tan x}{1+tan^{2}x}$
=$2 \frac{sin x}{cos x}\times\frac{1}{1+\frac{sin^{2}x}{cos^{2}x}}$
=$2 \frac{sin x}{cos x}\times\frac{cos^{2}x}{cos^{2}x + sin^{2}x}$
=$2 sin x \times \frac{cos x}{cos^{2}x + sin^{2}x}$
=$2 sin x \times \frac{cos x}{1}$
=$2 sin x cos x$
=$sin2x$
left=right
so, proved